Newton's Third Law Calculator

Enter the action force one object exerts on another (with each object's mass) to get the reaction force (F₁₂ = -F₂₁) and each object's resulting acceleration (a = F/m).

Quick Facts

Newton's Third Law
F(A on B) = -F(B on A)
Every action force is matched by a reaction force of equal magnitude, opposite direction, acting on the other object.
Forces act on different bodies
They never cancel
Action and reaction forces act on two separate objects, so they can't cancel each other out — only forces on the same object can.
Newton's Second Law
a = F / m
Equal forces still produce different accelerations when the two objects have different masses.

Your Results

Calculated
Reaction Force Magnitude
-
Force exerted by B back on A — equal to the action force
Reaction Force Direction
-
Action angle + 180°, opposite the action force
Acceleration of Object A
-
From the reaction force: a = F / mass A
Acceleration of Object B
-
From the action force: a = F / mass B

Ready

Enter the action force, its direction, and both masses, then press Calculate.

Formula and Method for Newton's Third Law

Newton's Third Law of Motion states that forces always come in pairs: when Object A exerts a force on Object B, Object B simultaneously exerts a force of equal magnitude but opposite direction back on Object A. As an equation, F(A on B) = -F(B on A). This calculator takes the action force between two objects — its magnitude, direction, and the mass of each object — and returns the reaction force, plus, using Newton's Second Law (a = F/m), the acceleration each object experiences from its half of the pair.

How the reaction force is calculated

Enter the magnitude and direction of the action force that Object A exerts on Object B. The calculator reports the reaction force that Object B exerts back on Object A: the same magnitude, in the opposite direction (the input angle plus 180°, wrapped to 0-360°). This equality is exact — it does not depend on the masses, speeds, or shapes of the two objects, and it holds for every interacting pair, from planets orbiting each other to two colliding billiard balls.

Why equal forces don't mean equal motion

Newton's third law only fixes the forces; it says nothing about how much each object moves. That comes from Newton's second law, F = ma, rearranged to a = F/m. Because the action and reaction forces share the same magnitude F, the object with the smaller mass accelerates more. With the default example — two ice skaters (60 kg and 90 kg) pushing off each other with 300 N — the lighter skater accelerates at 5 m/s² while the heavier skater accelerates at only 3.33 m/s², even though both feel exactly the same size force.

Common mistakes

  • Thinking action-reaction forces cancel: they can't, because they act on two different objects. Forces only cancel when both act on the same object.
  • Confusing the third law with equilibrium: an object at rest is in equilibrium because the forces acting on it (often from several different sources) sum to zero — that's a separate idea from the third law's action-reaction pairing.
  • Forgetting mass when predicting motion: equal-and-opposite forces do not produce equal accelerations unless the two masses are also equal.

Frequently Asked Questions

What is Newton's Third Law of Motion?
Newton's Third Law states that when one object exerts a force on a second object, the second object simultaneously exerts a force of equal magnitude and opposite direction back on the first: F(A on B) = -F(B on A). These paired forces are called an action-reaction pair.
Why don't action and reaction forces cancel each other out?
Because they act on two different objects, not the same one. Forces only cancel when both act on the same body. In an action-reaction pair, one force acts on Object A and the other acts on Object B, so each object still feels its own net force.
If the forces are equal, why don't both objects move the same way?
Newton's third law only fixes the forces as equal and opposite; it says nothing about motion. Motion comes from Newton's second law, a = F/m. Since the two objects usually have different masses, the same force magnitude produces different accelerations — the lighter object accelerates more.
What are real-world examples of Newton's Third Law?
Rocket propulsion (exhaust pushed backward, rocket pushed forward), walking (foot pushes the ground backward, ground pushes the foot forward), swimming, and firearm recoil (gun pushes the bullet forward, bullet pushes the gun backward) are all action-reaction pairs.

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